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- #[[MA284 - Discrete Mathematics]]
- **Previous Topic:** [[Trees]]
- **Next Topic:**
- **Relevant Slides:** ![MA284-Week12.pdf](../assets/MA284-Week12_1669208668908_0.pdf)
-
- In a practical setting, a graph must be stored in some computer-readable format.
- One of the most common is an **adjacency matrix**.
-
- # Adjacency Matrices
- What is an **adjacency matrix**? #card
- If the graph has $n$ vertices, labelled $\{1,2,\cdots, n\}$, then the **adjacency matrix** is an $m \times n$ **binary** matrix, $A$, with entries
- $$a_{i,j} =
\begin{cases}
1 & \text{vertex } i \text{ is adjacent to } j\\
0 & \text{otherwise.}
\end{cases}$$
- ![image.png](../assets/image_1669209153547_0.png)
- ## Properties of the Adjacency Matrix #card
- The adjacency matrix of a graph is **symmetric**.
- If $B = A^k$, then $b_{i,j}$ is the number of paths of length $k$ from vertex $i$ to vertex $j$.
- We can work out if a graph is connected by looking at the eigenvalues of $A$.
- If the graphs $G$ & $H$ are isomorphic, and have adjacency matrices $A_G$ & $A_H$, then there is a permutation matrix $P$, such that $PA_GP^{-1}=A_H$.
- The adjacency matrix idea is easily extended to allow for multigraphs and pseudographs (graphs with loops).
- For a multigraph, $a_{i,j}$ is the number of edges joining vertices $i$ & $j$.
- For a pseudograph, $a_{i,i}$ means that there is an edge from the vertex $i$ to itself.
- # Instance Matrices
- Graphs can also be represented by an **Incidence Matrix**.
- If the graph has $v$ vertices, and $e$ edges, then it is an $v \times e$ binary matrix.
- The rows represent vertices.
- The columns represent edges.
- If the matrix is $B = (b_{i,j})$ then $b_{ik} = 1$ means that the vertex $i$ is incident to edge $j$.
-
- # Distance Matrices
- What is the **eccentricity of a vertex**?
- The **eccentricity of a vertex** is the greatest distance between that vertex & any other vertex in the graph.
- What is the **radius of a graph**? #card
- The **radius of a graph** is the minimum eccentricity of any vertex.
- What is the **diameter of a graph**?
- The **diameter of a graph** is the maximum eccentricity of any vertex.
- This is also the maximum entry in the distance matrix.
-
-